where $u_n=u_n(t)$ is the displacement of string $n$ of eigen-frequency $f_n$ at time $t$ and the dot represents time differentiation, $U$ is the displacement of the soundboard with eigen-frequency $F$ and small damping coefficient $D$ representing outgoing sound, $u=\frac{1}{N}\sum_nu_n$ and the right hand side represents the connection between strings and soundboard through the bridge as a spring with spring constant $B$ (with $B\le F$ say). We consider a case of near-resonance with $f_n\approx F$ for $n=1,...,N$ (with a difference of about 1 Hz in a basic case with $F=440$ Hz say).

We are interested in the phase shift between $u_n$ and $U$ in the two basic cases: (i) zero phase shift with strings and soundboard moving together in "unison" mode and (ii) half period phase shift with strings and soundboard moving in opposition in "breathing" mode.

We have by summing over $n$ in (1), concentrating on the interaction between strings and soundboard thus omitting here the damping from outgoing sound setting $D=0$:

$\ddot u + F^2u+\frac{1}{N}\sum_n(f_n^2-F^2)u_n=B(U-u)$ (3)

Introducing $\phi = U+u$ and $\psi =U-u$ representing the two basic modes, we have by summing and subtracting (2) and (3):

$\ddot \phi + F^2\phi = -\frac{1}{N}\sum_n(f_n^2-F^2)u_n\approx 0$

$\ddot \psi + (F^2+2B)\psi =\frac{1}{N}\sum_n(f_n^2-F^2)u_n\approx 0$

with $\phi\approx 2U$ and $\psi\approx 0$ in case (i), and $\phi\approx 0$ and $\psi\approx 2U$ in case (ii), and $F^2+B\approx F^2$.

The difference between the two cases comes out in (1): In case (i) the average of $B(U-u_n)$ is small while in case (ii) the average of $B(U-u_n)\approx 2BU$. The right hand side $B(U-u_n)$ in (1) therefore acts to keep the different $u_n$ in-phase in case (ii), but does not exercise this stabilising effect in case (i), nor in the case of only one string.

The result is that the "breathing" mode of case (ii) can sustain a long aftersound with a sustained energy transfer from strings to soundboard until the strings and soundboard come to rest together.

On the other hand, in case (i) the strings will without the stabilising effect quickly go out of phase with the result that the energy transfer to the soundboard ceases and the outgoing sound dies while the strings are still oscillating, thus giving short aftersound.

You can follow these scenarios in case (i) here and in case (ii) here. We see 10 oscillating strings in blue and a common soundboard in red with strings in yellow, and staples showing string energy in blue and soundboard energy in red. We see strings and soundboard fading together in case (ii) with long aftersound, and soundboard fading before the strings in case (i) with short aftersound.

We saw in The Secret of the Piano 1 that the hammer initialises case (ii) and we have thus now uncovered the reason that there are 2-3 strings nearly equally tuned for most tones/keys of the piano: long aftersound with strings and soundboard fading slowly together.

More precisely, initialising the soundboard from rest by force interaction through the bridge with already initialised string oscillation, will in start-up have the soundboard lagging one quarter of period after the strings with corresponding quick energy transfer, and the phase shift will then tend to increase because the soundboard is dragged by the damping until the "breathing" mode with half period phase shift of case (ii) is reached with slower energy transfer and long aftersound. The "unison" mode with a full period (zero) phase shift will thus not be reached.

The analysis of the interaction string-soundboard may have relevance also for radiative interaction between different bodies as exposed on Computational Blackbody Radiation by suggesting an answer to the following question which has long puzzled me:

What coordinates the atomic oscillations underlying the radiation of a radiating body?

söndag 13 december 2015

The empty treaty (agreement) of COP21 without measurable reductions of CO2 emissions, saves humanity from the threatening disaster of a treaty asking for drastic reductions of CO2 emissions all the way to zero by 2050.

No wonder that the world leaders behind this monumental rescue operation raise there hands under massive ovations from all the people around the world, who can now go back to business.

One thing though: The treaty limits global warming to plus 1.5 C, but says nothing about global cooling. Hopefully that can be fixed in a revision: A limit to minus 0.5 C of global cooling would prevent another Ice Age, and stable global temperatures (and inflation of 2%) has shown to be the right climate for global business (and global happiness).

Another thing: The $100 billion dollars per year of transfer from rich to poor hinted at in the treaty, represents about 0.2% of the world GNP or 1% of China's GNP and thus like the warming of CO2 in the atmosphere, has a non-measurable empty effect.

fredag 11 december 2015

Negotiators at the Paris summit aim to wrap up a global agreement to curb climate change on Saturday - a day later than expected.

"We are nearly there. I'm optimistic," said French Foreign Minister Laurent Fabius, who is chairing the summit.

Efforts to forge a deal faltered on Friday, forcing the talks to over-run.

UN Secretary General Ban Ki-moon said the negotiations were "most complicated, most difficult, but, most important for humanity".

Mr Fabius told reporters in Paris that he would present a new version of the draft text on Saturday morning at 0800 GMT, which he was "sure" would be approved and "a big step forward for humanity as a whole".

"We are almost at the end of the road and I am optimistic," he added.

The new version to be presented on Saturday morning by Mr Fabius, firmly determined to lead the 195 countries on Mother Earth "to do something" to save Mother Earth from humanity, has however leaked and shows a stunning complete U-turn, apparently resulting from a sudden insight in the brain of Mr Fabius that CO2 is beneficial to both Mother Earth and humanity and that it would be completely insane to lead the world into a fossil free Hell by 2050:

Summary Statement

Over thirty years of intense (and extremely expensive) research has totally failed to produce any evidence that human emissions of CO2 are driving climate. CO2 is not a danger to but a benefit for all life on our planet.

We call on governments, NGOs and universities to stop pursuing policy and dogma based ‘evidence’ gathering.

• That they stop scaremongering.

• That they dissolve the IPCC and the UNFCCC.

• That governments focus instead on encouraging means of ensuring that under-developed and developing nations have full access to the cheapest reliable energy (particularly electricity), regardless of whether fossil fuels are used, so as to improve their access to clean water, low pollution cooking facilities and good medical services.

• That once respected academic institutions and scientific publications put their own houses in order and once again allow the free exchange of scientific ideas and results without prejudice.

• That those involved in alleged cases of scientific fraud, which have resulted in huge financial costs, causing greater poverty and many deaths among the poorest, be brought before the relevant Court of Law.10th Dec 2015Further, more detailed statements, references and videos of presentations at the Conference are available on the website: www.pcc15.org

onsdag 9 december 2015

We continue our study of the interaction of string and soundboard through a bridge of a piano with focus on the initialisation where the string is hit by the hammer and through the bridge transfers energy to the soundboard.

We model the string-soundboard-bridge system by the following coupled wave equations: Find functions $u(x,t)$ and $U(x,t)$ representing displacements of string and soundboard from initial flat configuration, such that for $0\lt x\lt 1$ and $t\gt 0$

where $S$ is a spring constant representing a springy connection of string and soundboard through a bridge located at $x=1$.

We initialise by setting $u(x,0)=U(x,0)=0$, $\frac{\partial u}{\partial t}(x,0)=1$ for $0.4\lt x\lt 0.6$ and $\frac{\partial u}{\partial t}(x,0)=0$ else, and $\frac{\partial U}{\partial t}(x,0)=0$ for $0\lt x\lt 1$, corresponding to hitting the string with the hammer, and watch the result here with the string red and soundboard blue.

We compare with initial data for $u$ changed to $u(x,0)=\sin(\pi x)$ and $\frac{\partial u}{\partial t}(x,0)=0$ with somewhat different response here.

We see that the motion settles into periodic modes of string and soundboard with a phase difference of half a period with the bridge basically at rest: when the string deflects upward the soundboard deflects downward and vice versa with zero net force on the bridge.

We have thus recovered the "breathing" motion with the bridge at rest of the previous post as the basic resonance mechanism of a piano allowing long aftersound with slow transfer of energy from string to soundboard.

The secret is hidden in the following dynamic wave model representing an instrument composed of $N$ strings connected to a common soundboard by a common bridge: For $n=1,..,N,$ and $t>0$

$\ddot u_n + f_n^2u_n=B(U-u_n)$

$\ddot U + F^2U+D\dot U=B(u_n-U)$

where $u_n=u_n(t)$ is the displacement of string $n$ of eigen-frequency $f_n$ at time $t$ and the dot represents time differentiation, $U$ is the displacement of the soundboard with eigen-frequency $F$ and damping coefficient $D$ representing outgoing sound, and the right hand side represents the connection between strings and soundboard through the bridge as a spring with spring constant $B$. We consider a case of near-resonance with $f_n\approx F$ for $n=1,...,N$, with a difference of about 1 Hz in a basic case with $F=440$ Hz say.

We can think of this model as composed of $N+1$ masses each connected to a fixed support by elastic springs ($N$ strings and 1 common soundboard ) joined by elastic springs connecting each string to the common soundboard/bridge through an elastic spring.

Recall that for a piano up to three strings are used for each single tone.

The performance of the instruments is expressed by the following energy balance obtained by multiplying 1. by $\dot u_n$ and 2. by $\dot U$:

where $E=E(t)$ is the total energy of the instrument at time $t$ as the sum of the string energy, soundboard energy and "bridge energy" $\frac{1}{2}\sum_n(u_n-U)^2$.

A tone is initialised by setting the strings in motion by plucking (guitar), by bow (violin) or hammer (piano) and we now focus on the interaction of the strings and soundboard after initialisation as a sound is generated from the vibration of the soundboard into the surrounding air. In a subsequent post we will consider the initialisation with near-resonance as one key to the secret.

The key to the secret of the sound production is revealed by the following observation:

The displacements of strings and displacement of soundboard is maintained with a phase shift of one half period through interaction via the common bridge, although the eigen-frequenices of the strings are not exactly equal to the eigen-frequency of the soundboard.

In other words, the strings and soundboard vibrate in coordinated motion with maximal mutual displacement $(U-u_n) with strings moving up/down when soundboard is moving down/up in a "pumping motion" and thus with substantial bridge energy.

In the real case of a guitar, violin or piano, the pumping motion with substantial force exchange between string and soundboard, is reflected by zero motion of the bridge with string and sound board pulling in opposite directions.

The secret of the sound production of the instrument is hidden in the following question:

What sustains sound production by coordinated string-soundboard motion with all strings with a half-period phase shift with a string-soundboard eigenfrequency difference of 1 Hz?

The answer comes out by subtracting 1. and 2. to get for $w_n=U-u_n$ for $n=1,...,N$

$\ddot w_n+\tilde F^2w_n\approx 0$,

where $\tilde F^2\approx F^2+2B\approx F^2$ if $B\le F$. The difference $U-u_n$ thus comes out as the same eigen-function for all $n$ with the phase shift of all strings coordinated to a common half-period phase shift vs the soundboard.

On the other hand, adding 1. and 2. gives for v_n=U+v_n

$\ddot v_n+F^2v_n\approx 0$,

as an eigen-function of frequency $F$ with $F^2<\tilde F^2$, representing motion with $u_n$ in-phase with $U$.

It then remains to explain why the mode $w_n$ with half-period phase shift and substantial bridge force is preferred by the instrument before the mode $v_n$ with a full period (or zero) phase shift and zero bridge force. I will return to this question in the next post starting with a study of the initialisation dynamics.

The model tells in the half period phase shift case that the sound dies quickly as soon as the strings are damped, because that means that both the string energy and the bridge energy is put to zero leaving only a the minor portion of soundboard energy for continued sound production.

onsdag 25 november 2015

Computation Blackbody Radiation presentats a new approach to Planck's radiation law based on finite precision computation applied to a wave model consisting a set of harmonic oscillators with small damping subject to near-resonant forcing, each one of the oscillators of the form

$\ddot u(t)+\nu^2u(t)+\gamma\dot u(t)=f(t)$ for time $t>0$,

where $\dot u=\frac{du}{dt}$ and $\ddot u=\frac{d^2u}{dt^2}$, $\nu >> 1$ is the eigenfrequency of the oscillator, $\gamma $ is a small positive damping coefficient with $\gamma\nu\le 1$, and $f(t)$ is a near-resonant forcing, for example given by

$f(t)=\sin((\nu -0.5)t)+\sin(\nu t)+\sin((\nu +0.5)t)$

with a total frequency shift of 1 Hz.

A basic aspect of this model connects to the so called Italian tuning of a piano, where the three strings in the middle high register for each key/tone are tuned with a total frequency shift of about 1 Hz as above.

Let us now seek to understand in what sense Italian tuning is different from standard tuning with the three strings for each key/tone tuned to exactly the same frequency or pitch. We then identify in the above model

the three strings are represented by the forcing $f(t)$,

the sound board of the piano is represented by the oscillator,

the outgoing sound from the sound board is represented by the damping.

Let us first consider standard tuning with strings and soundboard all with the same eigen-frequency, that is a case with perfect resonance. A pressed key activates a hammer with hits the strings which start to vibrate and thereby as forcing transfers energy to the sound board, which in turn starts to vibrate and produces a sound. In this case the forcing $f(t)$ will stay in phase with the velocity $\dot u$ over time, which means quick transfer of energy from strings to sound board as the integral of the positive product $f(t)\dot u(t)$ as the work performed per unit time. The result may be an outgoing sound of relatively high volume but short duration.

On the other hand, with the Italian tuning, the forcing from each of the three strings with slightly different frequencies cannot all be in phase with the common sound board velocity $\dot u$ over time, which means less quick energy transfer to the sound board with $f(t)\dot u(t)$ of changing sign and thus slower string energy loss as compared to the standard case. The result may be a sound of less volume but longer duration (sustain) than in the standard case, and also with slight "beat".

Of course, in reality it may be difficult to clearly separate the two cases, because perfect resonance does not really occur for a real piano with standard tuning, because of the complexity of the sound board, and also because the damper stops the string vibration before the tone has faded.

In any case, the distinction between perfect and near resonance is fundamental in the new proof of Planck's radiation law offered as by Computation Blackbody Radiation, a proof without reference to mystical statistics.

PS It is possible to change the setting by letting $f(t)$ represent the outgoing sound and the damping the input from the strings to the sound board. This is of relevance in stationary periodic state with sustained sound over long time without damping.

New Theory of Flight springs from our resolution of d'Alembert's paradox formulated in 1755, published in JMFM 2008, and shows for the first time that both simulation and understanding of the miracle of flight is possible with present computer and brain power. As such it has every chance of becoming a landmark article, as expressed by the Editorial Foreword:

The special character of this article requires some comments by the editors on the purpose of its publication. ...the implications of the argument and the accompanying novel numerical computations are of such far reaching importance for technical fluid dynamics, particularly for the computation of certain features in turbulent flow, that it deserves serious consideration. The main purpose of this publication is therefore to stimulate critical discussion among the experts in this area about the relevance and justification of the view taken in this article and its possible consequences for modeling and computation of turbulent flow.

fredag 23 oktober 2015

I have long expected to see a connection between jazz improvisation and computer games as a form of interactive computation where each player interacts with the other players as the music/computation evolves. Viewing music as a form of computation, jazz improvisation thus corresponds to interactive computation and here a jazz playing robot poses an interesting challenge to interactive computation or human-computer interaction.

onsdag 14 oktober 2015

an international panel of judges is planning to prohibit talk of “climate skepticism.” The article by Christopher Booker said the legal change was endorsed by Prince Charles of the U.K., and would revolutionize the concept of free speech once and for all.

Since it is now unlikely that the world will agree in Paris to a legally binding treaty to limit the rise in global temperatures to no more than 2 degrees C from pre-industrial levels, it is now time for the courts to step in, to enforce this as worldwide law.

söndag 11 oktober 2015

Luciano Floridis concludes in The Philosophy of Information that the world on microscopic scales cannot be neither discrete nor continuous, since each of these standpoints is contradictory, but can maybe instead be described by Informational Structural Realism (ISR) as scale dependent structural description leaving out the true nature of the elements forming the structure. ISR can be viewed as synthesis of the discrete and continuous without internal contradiction and thus potentially as a useful world view.

This is nothing but the finite element method in multi-scale form, originally developed in structural mechanics, as scale dependent discretization of continuous differential equations, where a true physical realisation of the finite elements is not possible nor necessary.

This connects to my attempt to describe a complex world including turbulence and quantum mechanics as analog finite precision computation simulated by digital finite precision computation, where the flow of information under computation represents ISR and the true physical nature of the analog computation is unknowable, but irrelevant.

Why is then both discrete and continuous physics impossible? Because both requires infinite resolution: a discrete point particle or discontinuity has zero size and a continuum has no smallest size. Thus both discrete and continuous physics requires infinitely small resolution and thus an infinite amount of information on any scale. If you don´t think this is asking for too much, then you should reconsider your notion of the infinite.

fredag 2 oktober 2015

There is no known physics theory that is true at every scale—there may never be.

When quantum mechanics is combined with relativity, it turns out, rather unexpectedly in fact, that the detailed nature of the physical laws that govern matter and energy actually depend on the physical scale at which you measure them.

So, what is going on? Is a universal theory a legitimate goal, or will scientific truth always be scale-dependent?

The trouble Krauss is here talking about is that relativity theory and quantum mechanics are contradictory, a deep trouble which has brought modern physics into a deep crisis. Krauss asks if a universal theory as a theory somehow without contradictions, is thinkable and then crushes all hope that it could take the form of string theory:

There is no example so far where an extrapolation as grand as that associated with string theory, not grounded by direct experimental or observational results, has provided a successful model of nature. In addition, the more we learn about string theory, the more complicated it appears to be, and many early expectations about its universalism may have been optimistic.

Krauss concludes with the pessimistic message that a universal theory as a theory without internal contradictions, is beyond reach for the human mind. The logic is that since both relativity and quantum theory are correct but unfortunately contradictory, no universal theory theory without contradiction is possible.

But why is it not thinkable that relativity or quantum theory is not correct physics? My bet is relativity theory is incorrect physics, supported by in particular the fact that while quantum mechanics has been awarded a countless number of Nobel Prizes, none has been awarded to relativity theory, of course because nobody in the Nobel committee could ever understand anything of Einstein's curved space-time.

Another obstacle to a universal theory is that the nature of quantum mechanics is postulated to be fundamentally different from the continuum mechanics of macroscopic phenomena and as such is beyond description in terms of concepts understandable to the human mind such as the partial differential equations making continuum mechanics understandable. But this is an ad hoc postulate blocking progress. How come that physicists can be so sure that the world of atoms is beyond human comprehension as expressed by Richard Feynman:

I think I can safely say that nobody understands quantum mechanics.

So, after all a TOE may be thinkable, if only we do not limit the thinking by relativity and quantum theory beyond reach for the human mind.

It is maybe not necessary as scientist to be paralyzed by ideas and concepts beyond human understanding as characteristics of religion.

lördag 26 september 2015

Germany with Angela Merkel is actively seeking to take the leading role in a giant transformation of the world economy into a new green economy with reductions of CO2 emissions to preindustrial levels as prime goal. In this giant transformation German car industry has promoted the diesel engine as being more fuel efficient with less CO2 emission than the gasoline engine under strong support from German governmental political correctness.

The Volkswagen emission scandal shows the hollowness and hypocrisy of this grand scale religion: To meet the strict demands of political correctness and moral leadership set by Germany, grand scale cheating is necessary and is accordingly delivered by Germany.

The world is watching with amazement. And in China a new coal power plant is opened every day.

torsdag 24 september 2015

Consider then a neutral atom of kernel charge $Z$ with $N=Z$ electrons occupying non-overlapping domains in space. Assume that the electrons are partitioned into a sequence of shells $S_m$ of increasing radius $r_m$ with corresponding widths $d_m$ each shell being filled by $2m^2$ electrons, for $m=1,...,M,$ with $M$ the number of shells. We consider a hypothetic atom with all shells fully filled with $2, 8, 18, 32, 50,...,$ electrons in successive shells displaying a basic aspect of the periodicity of the periodic table of elements.

Consider now the case $d_m\sim m$ with $r_m\sim m^2$, and assume $r_1=d_1\sim\frac{1}{Z}$. The electron density $\rho_m$ in $S_m$, assumed to be spherically symmetric, then satisfies

$\rho_mr_m^2d_m\sim m^2$

from which follows that

$\rho_m\sim \frac{m^3}{r_m^3}$. (1)

We now compute the following characteristics of this model:

$M^3\sim Z$, that is $M\sim Z^{\frac{1}{3}}$

potential energy in $S_1\sim Z^2$

potential energy in $S_m\sim m^2Z/r_m\sim Z/d_1\sim Z^2$

total potential energy and thus total energy $\sim Z^{\frac{7}{3}}$. (2)

We check that indeed there is room for $m^2$ electrons in shell $S_m$, because the volume of $S_m$ is $r_m^2d_m\sim m^5$, while the volume of an electron $\sim d_m^3\sim m^3$.

We observe that (2) fits with observations. We understand that the electronic density is distributed so that the potential energy and thus total energy in each full shell is basically the same, which may be viewed to be a heavenly socialistic organization of the shell structure of an atom.

Numerical computation seeking the ground state energy by relaxation in the Schrödinger model of post 5 starting from an initial density distribution according to (1), shows good correspondence with observation, supporting the basic analysis of this post. Numbers will be presented in an upcoming post.

The basic aspect of this model as a form of electron density model, is that electrons (or shells in the present spherically symmetric case) keep individuality by occupying different domains of space, which makes it possible to accurately represent electron-electron repulsion.

This feature is not present in standard density models such as Thomas-Fermi and Density Functional Theory. In these models electrons lack individuality as parts of electron clouds, which makes it difficult to represent electron-electron repulsion ab ibnitio.

Recall also that in the standard Schrödinger equations wave functions appear as multi-dimensional linear combinations of products of one-electron wave functions defined in all of space by separate spatial variables, thus with each electron "both nowhere and everywhere" without individuality, which requires a statistical interpretation of the wave function as a multi-dimensional uncomputable monster.

Another basic aspect of the presented model is continuity of electron density across inter-electron or inter-shell boundaries for the electron configuration of ground states. This allows atoms to have stable ground states as non-dissipative periodic states of minimal energy.

Notice further that the size of the atom as $r_M\sim Z^{-\frac{1}{3}}$ with decreasing size as $Z$ increases, corresponds to the observed decrease of size moving to the right in each row of the periodic table:

First of all, space-time is not a fabric. Space and time are not tangible 'things' in the same way that water and air are. It is incorrect to think of them as a 'medium' at all.

No physicist or astronomer versed in these issues considers space-time to be a truly physical medium, however, that is the way in which our minds prefer to conceptualize this concept, and has done so since the 19th century.

We really do not know what space-time is, other than two clues afforded by quantum mechanics and general relativity.

Space-time does not claim existence in its own right, but only as a structural quality of the [gravitational] field. (Einstein)

Space and time coordinates are just four out of many degrees of freedom we need, to specify a self-consistent theory. What we are going to have [in any future Theory of Everything] is not so much a new view of space and time, but a de-emphasis of space and time. (Steven Weinberg)

In the theory of gravity, you can't really separate the structure of space and time from the particles which are associated with the force of gravity [ such as gravitons]. The notion of a string is inseparable from the space and time in which it moves.(Michael Greene)

The punch line of this educational experience is presented in this way:

So, the question about what happens to space-time when a particle moves through it at near the speed of light is answered by saying that this is the wrong question to ask. Just because the brain can construct a question doesn't mean that the question has a physical answer!

We understand that LIGO in its search for "ripples in the fabric of space and time" is studying "the wrong question" and thus can be viewed as a study into the ""fabric of fantasy" which has become such a fundamental part of modern physics demanding full devotion by the sharpest brains of modern physicists (see also here ):

torsdag 17 september 2015

The LIGO scientific and engineering team at Caltech and MIT has been leading the effort over the past seven years to build Advanced LIGO, the world's most sensitive gravitational-wave detector.

Gravitational waves were predicted by Albert Einstein in 1916 as a consequence of his general theory of relativity, and are emitted by violent events in the universe such as exploding stars and colliding black holes.

Experimental attempts to find gravitational waves have been on going for over 50 years, and they haven't yet been found. They're both very rare and possess signal amplitudes that are exquisitely tiny.

Although earlier LIGO runs revealed no detections, Advanced LIGO, also funded by the NSF, increases the sensitivity of the observatories by a factor of 10, resulting in a thousandfold increase in observable candidate objects.

The original configuration of LIGO was sensitive enough to detect a change in the lengths of the 4-kilometer arms by a distance one-thousandth the diameter of a proton; this is like accurately measuring the distance from Earth to the nearest star—over four light-years—to within the width of a human hair.

Advanced LIGO, which will utilize the infrastructure of LIGO, is much more powerful.

The improved instruments will be able to look at the last minutes of the life of pairs of massive black holes as they spiral closer together, coalesce into one larger black hole, and then vibrate much like two soap bubbles becoming one.

In addition, Advanced LIGO will be used to search for the gravitational cosmic background, allowing tests of theories about the development of the universe only $10^{-35}$ seconds after the Big Bang.

Read these numbers: The accuracy of old LIGO was

the diameter of a human hair over a distance of 4 light-years,

$10^{-35}$ seconds after Big Bang,

and yet not the slightest little gravitational wave signal was recorded from even the most violent large scale phenomena thinkable. The conclusion should be clear: There are no gravitational waves. After all, why should there be any? By Einstein's general relativity which nobody claims to grasp?

But this is not the way Big Physics works: The fact that nothing was found by the infinitely sensitive LIGO requires an even more infinitely sensitive Advanced LIGO at a cost of a half a billion to be built by eager physicists, and after Advanced LIGO has found nothing, funding for an Advanced Advanced LIGO will be requested and so on...but why are tax payers supplying this Big Money?

lördag 5 september 2015

Gerhard 't Hooft is one of the Nobel Laureates in Physics who is not happy with the present state of understanding of quantum mechanics and seeks to do something about it: Hooft starts out in Determinism beneath Quantum Mechanics with:

The need for an improved understanding of what Quantum Mechanics really is, needs
hardly be explained in this meeting.

My primary concern is that Quantum Mechanics,
in its present state, appears to be mysterious.

It should always be the scientists’ aim to
take away the mystery of things.

It is my suspicion that there should exist a quite logical
explanation for the fact that we need to describe probabilities in this world quantum
mechanically.

This explanation presumably can be found in the fabric of the Laws of
Physics at the Planck scale.

However, if our only problem with Quantum Mechanics were our desire to demystify it,
then one could bring forward that, as it stands, Quantum Mechanics works impeccably.

It
predicts the outcome of any conceivable experiment, apart from some random ingredient.
This randomness is perfect. There never has been any indication that there would be any
way to predict where in its quantum probability curve an event will actually be detected.

Why not be at peace with this situation?

One answer to this is Quantum Gravity. Attempts to reconcile General Relativity with
Quantum Mechanics lead to a jungle of complexity that is difficult or impossible to interpret physically. In a combined theory, we no longer see “states” that evolve with “time”,
we do not know how to identify the vacuum state, and so on.

What we need instead is a
unique theory that not only accounts for Quantum Mechanics together with General Relativity, but also explains for us how matter behaves.

We should find indications pointing
towards the correct unifying theory underlying the Standard Model, towards explanations
of the presumed occurrence of supersymmetry, as well as the mechanism(s) that break it.
We suspect that deeper insights in what and why Quantum Mechanics is, should help us
further to understand these issues.

Hooft thus acknowledges that quantum mechanics is mysterious, which all prominent physicists do, but Hooft is not at peace with this situation, since after all the essence of science is understanding, although most of his colleagues seem to have accepted once and for all that quantum mechanics cannot be understood and cannot be reconciled with general relativity.
Hooft then proceeds to seek a determinism behind quantum mechanics in the form of cellular automatons (also here).

I am pursuing another route to an understandable form of quantum mechanics as analog computation with finite precision, which in a way connects to Hooft's cellular automaton's, but is expressed by Schrödinger type wave equations in a continuum mechanics framework.

In this framework the finite precision computation makes a difference between smooth (strong) solutions and non-smooth (weak) solutions of the wave equations: Smooth solutions satisfy the wave equations exactly (with infinite precision), while non-smooth solutions satisfy the equations only in a weak sense with finite precision and loss of information as a form of dissipative radiation.

This allows the ground state of an atom as a smooth solution without dissipation to be stable over time without dissipation, while an excited state as a non-smooth solution will return to the ground state under dissipative radiation.

The situation is analogous to that described in my work together with Johan Hoffman on fluid mechanics, with turbulent solutions as non-smooth dissipative solutions of formally inviscid Euler equations, which allowed us to resolve d'Alembert's paradox (J Math Fluid Mech 2008) and formulate a new theory of flight (to appear in J Math Fluid Mech 2015), among other things.

onsdag 2 september 2015

The new Schrödinger equation I am studying in this sequence of posts takes the following form, in spherical coordinates with radial coordinate $r\ge 0$ in the case of spherical symmetry, for an atom with kernel of charge $Z$ at $r=0$ with $N\le Z$ electrons of unit charge distributed in a sequence of non-overlapping spherical shells $S_1,...,S_M$ separated by spherical surfaces of radii $0=r_0<r_1<r_2<...<r_M=\infty$, with $N_j>0$ electrons in shell $S_j$ corresponding to the interval $(r_{j-1},r_j)$ for $j=1,...,M,$ and $\sum_j N_j = N$:

Find a complex-valued differentiable function $\psi (r,t)$ depending on $r≥0$ and time $t$, satisfying for $r>0$ and all $t$,

$i\dot\psi (r,t) + H(r,t)\psi (r,t) = 0$ (1)

where $\dot\psi = \frac{\partial\psi}{\partial t}$ and $H(r,t)$ is the Hamiltonian defined by

Here $-\frac{Z}{r}$ is the kernel-electron attractive potential and $V(r,t)$ is the electron-electron repulsive potential computed using the fact that the potential $W(s)$ of a spherical uniform surface charge distribution of radius $r$ centered at $0$ of total charge $Q$, is given by $W(s)=Q\min(\frac{1}{r},\frac{1}{s})$, with a reduction for a lack of self-repulsion within each shell given by the factor $(N_j -1)/N_j$.

The $N_j$ electrons in shell $S_j$ are thus homogenised into a spherically symmetric charge distribution of total charge $N_j$.

This is a free boundary problem readily computable on a laptop, with the $r_j$ representing the free boundary separating shells of spherically symmetric charge distribution of intensity $\vert\psi (r,t)\vert^2$ and a free boundary condition asking continuity and differentiability of $\psi (r,t)$.

Separating $\psi =\Psi +i\Phi$ into real part $\Psi$ and imaginary part $\Phi$, (1) can be solved by explicit time stepping with (sufficiently small) time step $k>0$ and given initial condition (e.g. as ground state):

$\Psi^{n+1}=\Psi^n-kH\Phi^n$,

$\Phi^{n+1}=\Phi^n+kH\Psi^n$,

for $n=0,1,2,...,$ where $\Psi^n(r)=\Psi (r,nk)$ and $\Phi^n(r)=\Phi (r,nk)$, while stationary ground states can be solved by the iteration

$\Psi^{n+1}=\Psi^n-kH\Psi^n$,

$\Phi^{n+1}=\Phi^n-kH\Phi^n$,

while maintaining (2).

A remarkable fact is that this model appears to give ground state energies as minimal eigenvalues of the Hamiltonian for both ions and atoms for any $Z$ and $N$ within a percent or so, or alternatively ground state frequencies from direct solution in time dependent form. Next I will compute excited states and transitions between excited states under exterior forcing.

Specifically, what I hope to demonstrate is that the model can explain the periods of the periodic table corresponding to the following sequence of numbers of electrons in shells of increasing radii: 2, (2, 8), (2, 8, 8), (2, 8, 18, 8), (2, 8, 18, 18, 8)... which to be true lacks convincing explanation in standard quantum mechanics (according to E. Serri among many others).

The basic idea is thus to represent the total wave function $\psi (r,t)$ as a sum of shell wave functions
with non-overlapping supports in the different in shells requiring $\psi (r,t)$ and thus $\vert\psi (r,t)\vert^2$ to be continuous across inter-shell boundaries as free boundary condition, corresponding to continuity of charge distribution as a classical equilibrium condition.

I have also with encouraging results tested this model for $N\le 10$ in full 3d geometry without spherical shell homogenisation with a wave function as a sum of electronic wave functions with non-overlapping supports separated by a free boundary determined by continuity of wave function including charge distribution.

We compare with the standard (Hartree-Fock-Slater) Ansatz of quantum mechanics with a multi-dimensional wave function $\psi (x_1,...,x_N,t)$ depending on $N$ independent 3d coordinates $x_1,...,x_N,$ as a linear combination of wave functions of the multiplicative form

$\psi_1(x_1,t)\times\psi_2(x_2,t)\times ....\times\psi_N(x_N,t)$,

with each electronic wave function $\psi_j(x_j,t)$ with global support (non-zero in all of 3d space). Such multi-d wave functions with global support thus depend on $3N$ independent space coordinates and as such defy both direct physical interpretation and computability, as soon as $N>1$, say. One may argue that since such multi-d wave function cannot be computed, it does not matter that they have no physical meaning, but the net output appears to be nil, despite the declared immense success of standard quantum mechanics based on this Ansatz.

The information is not stored in the interior of the black hole as one might expect, but in its boundary — the event horizon,” he said. Working with Cambridge Professor Malcolm Perry (who spoke afterward) and Harvard Professor Andrew Stromberg, Hawking formulated the idea hat information is stored in the form of what are known as super translations.

The problem arises because quantum mechanics is viewed to be reversible, because the mathematical equations supposedly describing atomic physics formally are time reversible: a solution proceeding in forward time from an initial to a final state, can also be viewed as a solution in backward time from the earlier final state to the initial state. The information encoded in the initial state can thus, according to this formal argument, be recovered and thus is never lost. On the other hand a black hole is supposed to swallow and completely destroy anything it reaches and thus it appears that a black hole violates the postulated time reversibility of quantum mechanics and non-destruction of information.

Hawking's solution to this apparent paradox, is to claim that after all a black hole does not destroy information completely but "stores it on the boundary of the event horizon". Hawking thus "solves" the paradox by maintaining non-destruction of information and giving up complete black hole destruction of information.

The question Hawking seeks to answer is the same as the fundamental problem of classical physics which triggered the development of modern physics in the late 19th century with Boltzmann's "proof" of the 2nd law of thermodynamics: Newton's equations describing thermodynamics are formally reversible, but the 2nd law of thermodynamics states that real physics is not always reversible: Information can be inevitably lost as a system evolves towards thermodynamical equilibrium and then cannot be recovered. Time has a direction forward and cannot be reversed.

Boltzmann's "proof" was based an argument that things that do happen do that because they are "more probable" than things which do not happen. This deep insight opened the new physics of statistical mechanics from which quantum borrowed its statistical interpretation.

I have presented a different new resolution of the apparent paradox of irrreversible macrophysics based on reversible microphysics by viewing physics as analog computation with finite precision, on both macro- and microscales. A spin-off of this idea is a new resolution of d'Alemberts's paradox and a new theory of flight to be published shortly.

The basic idea here is thus to replace the formal infinite precision of both classical and quantum mechanics, which leads to paradoxes without satisfactory solution, with realistic finite precision which allows the paradoxes to be resolved in a natural way without resort to unphysical statistics. See the listed categories for lots of information about this novel idea.

The result is that reversible infinite precision quantum mechanics is fiction without physical realization, and that irreversible finite precision quantum mechanics can be real physics and in this world of real physics information is irreversibly lost all the time even in the atomic world. Hawking's resolution is not convincing.

Here is the key observation explaining the occurrence of irreversibility in formally reversible systems modeled by formally non-dissipative partial differential equations such as the Euler equations for inviscid macroscopic fluid flow and the Schrödinger equations for atomic physics:

Smooth solutions are strong solutions in the sense of satisfying the equations pointwise with vanishing residual and as such are non-dissipative and reversible. But smooth solutions make break down into weak turbulent solutions, which are only solutions in weak approximate sense with pointwise large residuals and these solutions are dissipative and thus irreversible.

An atom can thus remain in a stable ground state over time corresponding to a smooth reversible non-dissipative solution, while an atom in an excited state may return to the ground state as a non-smooth solution under dissipation of energy in an irreversible process.

fredag 28 augusti 2015

I have tested the new atomic model described in a previous post in setting of spherical symmetry with electrons filling a sequence of non-overlapping spherical shells around a kernel. The electrons in each shell are homogenized to spherical symmetry which reduces the model to a 1d free boundary problem with the free boundary represented by the inter-shell spherical surfaces adjusted so that the combined wave function is continuous along with derivates across the boundary. The repulsion energy is computed so as to take into account that electrons are not subject to self-repulsion, by a corresponding reduction of the repulsion within a shell.

The remarkable feature of this atomic model, in the form of a 1d free boundary problem with continuity as free boundary condition and readily computable on a lap-top, is that computed ground state energies show to be surprisingly accurate (within 1%) for all atoms including ions (I have so far tested up to atomic number 54 and am now testing excited states).

Recall that the wave function $\psi (x,t)$ solving the free boundary problem, has the form

$\psi (x,t) =\psi_1(x,t)+\psi_2(x,t)+...+\psi_S(x,t)$ (1)

with $(x,t)$ a common space-time coordinate, where $S$ is the number of shells and $\psi_j(x,t)$ with support in shell $j$ is the wave function for the homogenized wave function for the electrons in shell $j$ with $\int\vert\psi_j(x,t)\vert^2\, dx$ equal to the number of electrons in shell $j$.

Note that the model can be used in time dependent form and then allows direct computation of vibrational frequencies, which is what can be observed.

Altogether, the model in spherical symmetric form indicates that the model captures essential features of the dynamics of an atom, and thus can useful in particular for studies of atoms subject to exterior forcing.

I have also tested the model without spherical homogenisation for atoms with up to 10 electrons, with similar results. In this case the the free boundary separates diffferent electrons (and not just shells of electrons) with again continuous charge distribution across the corresponding free boundary.

In this model electronic wave functions share a common space variable and have disjoint supports and can be given a classical direct physical interpretation as charge distribution. There is no need of any Pauli exclusion principle: Electrons simply occupy different regions of space and do not overlap, just as in a classical multi-species continuum model.

This is to be compared with standard quantum mechanics based on multidimensional wave functions $\psi (x_1,x_2,...,x_N,t)$ typically appearing as linear combinations of products of electronic wave functions

for an atom with $N$ electrons, each electronic wave function $\psi_j(x_j,t)$ being globally defined with its own independent space coordinate $x_j$. Such multidimensional wave functions can only be given statistical interpretation, which lacks direct physical meaning. In addition, Pauli's exclusion principle must be invoked and it should be remembered that Pauli himself did not like his principle since it was introduced ad hoc without any physical motivation, to save quantum mechanics from collapse from the very start...

More precisely, while (1) is perfectly reasonable from a classical continuum physics point of view, and as such is computable and useful, linear combination of (2) represent a monstrosity which is both uncomputable and unphysical and thus dangerous, but nevertheless is supposed to represent the greatest achievement of human intellect all times in the form of the so called modern physics of quantum mechanics.

How long will it take for reason and rationality to return to physics after the dark age of modern physics initiated in 1900 when Planck's "in a moment of despair" resorted to an ad hoc hypothesis of a smallest quantum of energy in order to avoid the "ultra-violet catastrophe" of radiation viewed to be impossible to avoid in classical continuum physics. But with physics as finite precision computation, which I am exploring, there is no catastrophe of any sort and Planck's sacrifice of rationality serves no purpose.

PS Here are the details of the spherical symmetric model starting from the following new formulation of a Schrödinger equation for an atom with $N$ electrons organised in spherical symmetric form into $S$ shells: Find a wave function

and the wave functions are normalised to correspond to unit charge of each electron:

$\int_{\Omega_j}\vert\psi_j(x,t)\vert^2 =1$ for all $t$ for $j=1,..,N$.

Assume the electrons fill a sequence of shells $S_k$ for $k=1,...,S$ centered at the atom kernel with $N_k$ electrons on shell $S_k$ and

$\int_{S_k}\vert\psi (x,t)\vert^2 =N_k$ for all $t$ for $k=1,..,S$,

$\sum_k^S N_k = N$.

The total wave function $\psi (x,t)$ is thus assumed to be continuously differentiable and the electronic potential of the Hamiltonian acting in $\Omega_j(t)$ is given as the attractive kernel potential together with the repulsive kernel potential resulting from the combined electronic charge distributions $\vert\psi_k\vert^2$ for $k\neq j$, with total electronic repulsion energy

Assume now that the electronic repulsion energy is approximately determined by homogenising the $N_k$ electronic wave function $\psi_j$ in each shell $S_k$ into a spherically symmetric "electron cloud" $\Psi_k(x)$ with corresponding potential $W_k(y)$ given by

and $R_k(x)=\frac{N_k-1}{N_k}$ for $x\in S_k$ is a reduction factor reflecting non self-repulsion of each electron (and $R_k=1$ else): Of the $N_k$ electrons in shell $S_k$, thus only $N_k-1$ electrons contribute to the value of potential in shell $S_k$ from the electrons in shell $S_k$. We here use the fact that the potential $W(x)$ of a uniform charge distribution on a spherical surface $\{y:\vert y\vert =r\}$ of radius $r$ of total charge $Q$, is equal to $Q/\vert x\vert$ for $\vert x\vert >r$ and $Q/r$ for $\vert x\vert <r$.

Our model then has spherical symmetry and is a 1d free boundary problem in the radius $r=\vert x\vert$ with the free boundary represented by the radii of the shells and the corresponding Hamiltonian is defined by the electronic potentials computed by spherical homogenisation in each shell. The free boundary is determined so that the combined wave function $\psi (x,t)$ is continuously differentiable across the free boundary.